Uniformity of the Möbius function in F q [t]

Transcription

1 Uniformity of the Möbius function in F q [t] Thái Hoàng Lê 1 & Pierre-Yves Bienvenu 2 1 University of Mississippi 2 University of Bristol January 13, 2018 Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

2 The Möbius randomness principle Recall { ( 1) k if n = p µ(n) = 1 p 2 p k is a product of k distinct primes, 0 if n is not squarefree. Thus the sequence {µ(n)} is 1, 1, 1, 0, 1, 1,.... Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

3 The Möbius randomness principle Recall { ( 1) k if n = p µ(n) = 1 p 2 p k is a product of k distinct primes, 0 if n is not squarefree. Thus the sequence {µ(n)} is 1, 1, 1, 0, 1, 1,.... The Möbius randomness principle states that µ is random-like, Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

4 The Möbius randomness principle Recall { ( 1) k if n = p µ(n) = 1 p 2 p k is a product of k distinct primes, 0 if n is not squarefree. Thus the sequence {µ(n)} is 1, 1, 1, 0, 1, 1,.... The Möbius randomness principle states that µ is random-like, i.e. for any bounded, simple or structured function F, we have N µ(n)f(n) = o(n). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

5 Examples: 1 If F(n) = 1, then PNT is equivalent to N is equivalent to for any ɛ > 0. N µ(n) = O ɛ ( N 1/2+ɛ) µ(n) = o(n) and RH Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

6 Examples: 1 If F(n) = 1, then PNT is equivalent to N is equivalent to for any ɛ > 0. N µ(n) = O ɛ ( N 1/2+ɛ) µ(n) = o(n) and RH 2 If F(n) is periodic with period q, then N µ(n)f(n) = o(n) is equivalent to PNT in arithmetic progressions. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

7 3 Sarnak s conjecture: If (X, T ) is a dynamical system (i.e. X is a compact metric space, T : X X is continuous) with topological entropy zero, then for any x X and f C(X), N µ(n)f (T n x) = o(n). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

8 3 Sarnak s conjecture: If (X, T ) is a dynamical system (i.e. X is a compact metric space, T : X X is continuous) with topological entropy zero, then for any x X and f C(X), N µ(n)f (T n x) = o(n). 4 (Kalai/Green 2011) If F can be computed by bounded depth circuits, then N µ(n)f (n) = o(n). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

9 3 Sarnak s conjecture: If (X, T ) is a dynamical system (i.e. X is a compact metric space, T : X X is continuous) with topological entropy zero, then for any x X and f C(X), N µ(n)f (T n x) = o(n). 4 (Kalai/Green 2011) If F can be computed by bounded depth circuits, then N µ(n)f (n) = o(n). 5 (Dartyge-Tenenbaum 2005) If s(n) is the sum of digits of n in base q, then for any α R/Z, Here e(x) = e 2πix. N µ(n)e(s(n)α) = o(n). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

10 Exponential sums Davenport (1937): for any A > 0, N µ(n)e(nα) A N log A N uniformly in α R/Z. The implied constant is ineffective. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

11 Exponential sums Davenport (1937): for any A > 0, N µ(n)e(nα) A N log A N uniformly in α R/Z. The implied constant is ineffective. Baker-Harman (1991), Montgomery-Vaughan (unpublished): Assuming GRH, we have N µ(n)e(nα) ɛ N 3/4+ɛ uniformly in α R/Z, for any ɛ > 0. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

12 Since 1 0 N µ(n)e(nα) we cannot do better than N 1/2. 2 N = µ(n) 2 N, Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

13 Since 1 0 N µ(n)e(nα) we cannot do better than N 1/2. 2 = N µ(n) 2 N, Green-Tao (2008, 2012): If F(n) is a nilsequence, then for any A > 0, N µ(n)f (n) A N log A N. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

14 Since 1 0 N µ(n)e(nα) we cannot do better than N 1/2. 2 = N µ(n) 2 N, Green-Tao (2008, 2012): If F(n) is a nilsequence, then for any A > 0, N µ(n)f (n) A N log A N. Nilsequences include F(n) = e ( αn 2 + βn ) and F(n) = e ( nα βn). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

15 Function field analogy Let F q be the finite field on q elements. F q [t] is similar to Z in many aspects (both are UFDs). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

16 Function field analogy Let F q be the finite field on q elements. F q [t] is similar to Z in many aspects (both are UFDs). For f F q [t], define ( 1) k if f = cp 1 P 2 P k, P i distinct monic irreducibles, µ(f ) = c F q, 0 if f is not squarefree. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

17 Function field analogy Let F q be the finite field on q elements. F q [t] is similar to Z in many aspects (both are UFDs). For f F q [t], define ( 1) k if f = cp 1 P 2 P k, P i distinct monic irreducibles, µ(f ) = c F q, 0 if f is not squarefree. RH is true in F q [t]: for n 2, deg f =n, f monic µ(f ) = 0. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

18 Function field analogy Let F q be the finite field on q elements. F q [t] is similar to Z in many aspects (both are UFDs). For f F q [t], define ( 1) k if f = cp 1 P 2 P k, P i distinct monic irreducibles, µ(f ) = c F q, 0 if f is not squarefree. RH is true in F q [t]: for n 2, deg f =n, f monic µ(f ) = 0. Furthermore, GRH is true in F q [t] (Weil 1948). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

19 The field of fractions F q (t) = {f /g : f, g F q [t], g 0} is the analog of Q. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

20 The field of fractions F q (t) = {f /g : f, g F q [t], g 0} is the analog of Q. Define f /g = q deg f deg g. The completion of F q (t) with respect to is (( )) { } 1 n F q = α = a i t i : n Z, a i F q for every i, t i= the set of formal Laurent series in 1 t. This is the analog of R. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

21 The field of fractions F q (t) = {f /g : f, g F q [t], g 0} is the analog of Q. Define f /g = q deg f deg g. The completion of F q (t) with respect to is (( )) { } 1 n F q = α = a i t i : n Z, a i F q for every i, t i= the set of formal Laurent series in 1 t. This is the analog of R. Define T = {α F q (( 1 t )) : α < 1}. This is the analog of R/Z. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

22 Fix e q : F q {z C : z = 1} to be an additive character of F q. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

23 Fix e q : F q {z C : z = 1} to be an additive character of F q. For α = n i= a it i F q (( 1 t )), define e(α) = e q (a 1 ). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

24 Fix e q : F q {z C : z = 1} to be an additive character of F q. For α = n i= a it i F q (( 1 t )), define e(α) = e q (a 1 ). Thus e : F q (( 1 t )) {z C : z = 1} is an additive character of F q (( 1 t )). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

25 Problems we want to solve Problem 1. Let k 1 and Φ(x) = α k x k + α k 1 x k α 1 x be a polynomial in F q (( 1 t ))[x]. Show that µ(f )e(φ(f )) = o(q n ) deg f <n uniformly in Φ of degree k. Better yet, (since we have GRH) obtain a bound O(q c k n ) for some constant c k < 1. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

26 Problems we want to solve Problem 1. Let k 1 and Φ(x) = α k x k + α k 1 x k α 1 x be a polynomial in F q (( 1 t ))[x]. Show that µ(f )e(φ(f )) = o(q n ) deg f <n uniformly in Φ of degree k. Better yet, (since we have GRH) obtain a bound O(q c k n ) for some constant c k < 1. There are differences between this and the integer case. In fact, when k p = char(f q ), even the Weyl sum deg f <n e(φ(f )) is not well understood, since the Weyl differencing process breaks down (k! = 0). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

27 Problem 2. Let k 1 and Q F q [x 0, x 1,..., x n 1 ] be a polynomial of degree k. Show that µ(f )e q (Q(f )) = o(q n ) deg f <n uniformly in Q of degree k. Here Q(f ) is Q evaluated at the coefficients of f. Better yet, (since we have GRH) obtain a bound O(q c k n ) for some constant c k < 1. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

28 Problem 2. Let k 1 and Q F q [x 0, x 1,..., x n 1 ] be a polynomial of degree k. Show that µ(f )e q (Q(f )) = o(q n ) deg f <n uniformly in Q of degree k. Here Q(f ) is Q evaluated at the coefficients of f. Better yet, (since we have GRH) obtain a bound O(q c k n ) for some constant c k < 1. Problem 2 is more general than Problem 1 since for any polynomial Φ(x), (Φ(f )) 1 is a polynomial in the coefficients of f (recall that e(α) = e q ((α) 1 )). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

29 The linear case When k = 1 then Problem 1 and Problem 2 are the same. This is because for any linear form l F q [x 0,..., x n 1 ], there is α T such that l(f ) = (αf ) 1 for any deg f < n. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

30 The linear case When k = 1 then Problem 1 and Problem 2 are the same. This is because for any linear form l F q [x 0,..., x n 1 ], there is α T such that l(f ) = (αf ) 1 for any deg f < n. Theorem For any ɛ > 0, we have uniformly in α T. deg f <n µ(f )e(αf ) ɛ,q q (3/4+ɛ)n Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

31 Recall Baker-Harman and Montgomery-Vaughan s bound in Z (under GRH) N µ(n)e(αn) N 3/4+ɛ. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

32 Recall Baker-Harman and Montgomery-Vaughan s bound in Z (under GRH) N µ(n)e(αn) N 3/4+ɛ. Porritt independently proved a similar result which gives the same exponent 3/4 + ɛ when q is sufficiently large in terms of ɛ. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

33 Recall Baker-Harman and Montgomery-Vaughan s bound in Z (under GRH) N µ(n)e(αn) N 3/4+ɛ. Porritt independently proved a similar result which gives the same exponent 3/4 + ɛ when q is sufficiently large in terms of ɛ. Our argument is different from the proof in Z in some respects. This is because the topologies of T and R/Z are different and there is no summation by parts in F q [t]. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

34 We use Hayes L-functions of arithmetically distributed relations, as opposed to Dirichlet L-functions. Let M be the set of monic polynomials in F q [t]. Define an equivalence relation f g (mod R l,q ) on M as follows f g (mod Q) and the first l + 1 coefficients of f and g are the same. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

35 We use Hayes L-functions of arithmetically distributed relations, as opposed to Dirichlet L-functions. Let M be the set of monic polynomials in F q [t]. Define an equivalence relation f g (mod R l,q ) on M as follows f g (mod Q) and the first l + 1 coefficients of f and g are the same. Then M/R l,q is a semigroup, G l,q := (M/R l,q ) is a group. If λ is a nontrivial character of G l,q, then L(s, λ) = f M λ(f ) f s satisfies GRH (Rhin 1972). (If l = 0 then λ is a Dirichlet character.) Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

36 Theorem Assuming a quantitative form of the bilinear Bogolyubov theorem and p > 2. Then for any A > 0 and quadratic polynomial Q in F q [x 0,..., x n 1 ], uniformly in Q. deg f <n µ(f )e(q(f )) q,a q n n A Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

37 The quadratic case Theorem There is a constant c < 1 such that µ(f )e(αf 2 + βf ) q q cn uniformly in α, β T. deg f <n Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

38 Ideas of proof The classical circle method deals with sums like deg f <n µ(f )e(αf ) or deg f <n µ(f )e(αf 2 + βf ). To estimate such sums we need to distinguish two cases. α is close to a rational with small denominator (α is in the major arcs): use our understanding of the distribution of irreducibles in congruence classes, i.e. L-functions. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

39 Ideas of proof The classical circle method deals with sums like deg f <n µ(f )e(αf ) or deg f <n µ(f )e(αf 2 + βf ). To estimate such sums we need to distinguish two cases. α is close to a rational with small denominator (α is in the major arcs): use our understanding of the distribution of irreducibles in congruence classes, i.e. L-functions. α is not in the major arcs (α is in the minor arcs): use Vaughan s identity to decompose the sum into Vinogradov s Type I/Type II sums. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

40 In the case of deg f <n µ(f )e q(φ(f )), where Φ(x) = x T Mx and M is a symmetric matrix, the major arcs and minor arcs correspond to low rank and high rank matrices M. This is because e q (x T Mx) qn rank(m)/2. x F q n Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

41 In the case of deg f <n µ(f )e q(φ(f )), where Φ(x) = x T Mx and M is a symmetric matrix, the major arcs and minor arcs correspond to low rank and high rank matrices M. This is because e q (x T Mx) qn rank(m)/2. x F q n The low rank can be easily reduced to the linear case. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

42 In the case of deg f <n µ(f )e q(φ(f )), where Φ(x) = x T Mx and M is a symmetric matrix, the major arcs and minor arcs correspond to low rank and high rank matrices M. This is because e q (x T Mx) qn rank(m)/2. x F q n The low rank can be easily reduced to the linear case. Suppose for a contradiction that deg f <n µ(f )e q(φ(f )) δq n. We want to show that rank(m) is small. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

43 After using Vaughan s identity, Vinogradov s Type I/Type II sums, Cauchy-Schwarz and some combinatorial reasoning, we find that for some n k n, the set of pairs P h := {(a, b) : deg a, deg b G k+1 G k+1 : rank M a,b h} is large (has size (δ/n) O(1) q 2k+2 ) for some h = O(log(n/δ)). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

44 After using Vaughan s identity, Vinogradov s Type I/Type II sums, Cauchy-Schwarz and some combinatorial reasoning, we find that for some n k n, the set of pairs P h := {(a, b) : deg a, deg b G k+1 G k+1 : rank M a,b h} is large (has size (δ/n) O(1) q 2k+2 ) for some h = O(log(n/δ)). Here G m = {f : deg f < m}, M a,b = L T a ML b + L T b ML a and L a is the matrix of the map G n k G n, f af. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

45 Problem: We know P h := {(a, b) : deg a, deg b G k+1 G k+1 : rank M a,b h} is large, where M a,b = L T a ML b + L T b ML a. Show that rank M is small. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

46 Problem: We know P h := {(a, b) : deg a, deg b G k+1 G k+1 : rank M a,b h} is large, where M a,b = L T a ML b + L T b ML a. Show that rank M is small. If (a, b), (a, b) P h, then (a a, b) P 2h since M a a,b = M a,b M a,b. Similarly for the second coordinate. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

47 Problem: We know P h := {(a, b) : deg a, deg b G k+1 G k+1 : rank M a,b h} is large, where M a,b = L T a ML b + L T b ML a. Show that rank M is small. If (a, b), (a, b) P h, then (a a, b) P 2h since M a a,b = M a,b M a,b. Similarly for the second coordinate. By repeatedly smoothing in each coordinate, we find that P 64h contains an interesting structure. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

48 Theorem (Bilinear Bogolyubov Theorem) If P h ηq 2(k+1), then there exist F p -subspaces W 1, W 2 of the F p -vector space G k+1 of codimension r 1, r 2 and F p -bilinear forms Q 1,..., Q r on W 1 W 2 such that P 64h {(x, y) W 1 W 2 Q 1 (x, y) = = Q r (x, y) = 0} and max(r, r 1, r 2 ) c(η). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

49 Theorem (Bilinear Bogolyubov Theorem) If P h ηq 2(k+1), then there exist F p -subspaces W 1, W 2 of the F p -vector space G k+1 of codimension r 1, r 2 and F p -bilinear forms Q 1,..., Q r on W 1 W 2 such that P 64h {(x, y) W 1 W 2 Q 1 (x, y) = = Q r (x, y) = 0} and max(r, r 1, r 2 ) c(η). Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

50 Theorem (Bilinear Bogolyubov Theorem) If P h ηq 2(k+1), then there exist F p -subspaces W 1, W 2 of the F p -vector space G k+1 of codimension r 1, r 2 and F p -bilinear forms Q 1,..., Q r on W 1 W 2 such that P 64h {(x, y) W 1 W 2 Q 1 (x, y) = = Q r (x, y) = 0} and max(r, r 1, r 2 ) c(η). A similar result was proved independently by Gowers-Milicevic. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21

51 Theorem (Bilinear Bogolyubov Theorem) If P h ηq 2(k+1), then there exist F p -subspaces W 1, W 2 of the F p -vector space G k+1 of codimension r 1, r 2 and F p -bilinear forms Q 1,..., Q r on W 1 W 2 such that P 64h {(x, y) W 1 W 2 Q 1 (x, y) = = Q r (x, y) = 0} and max(r, r 1, r 2 ) c(η). A similar result was proved independently by Gowers-Milicevic. We can take c(η) = O(exp(exp(exp(log O(1) η 1 )))). This is too large to be useful. If we can take c(η) = log O(1) η 1 then in our application we can take δ = n A and this implies that M has small rank. Thái Hoàng Lê & Pierre-Yves Bienvenu Uniformity of the Möbius function JMM / 21